As I mentioned two days ago in my Tweetdeck and Facebook Connect Prompts posting getting TweetDeck to remember my Facebook credentials has been impossible.
Well turns out the issue stems from an un-announced/un-explained change that Facebook made to their API that was creating all the issues.
TweetDeck support has provided a page that explains what is happening and their solution which happens to be downloading an update for your TweetDeck software.
Most likely though you have already been prompted by TweetDeck to grab this update. When you start up TweetDeck you will see this prompt up in the title bar:
Just click that link and download the update for installation. As advertised the update fixed the issue.